To apply [1]}, Theorem 4.1, we need bounds for \(|\beta ^{(n)}(\xi )-b(x(\xi ))|\)  and \(\alpha ^{(n)}(\xi )\) , which we now derive. We partition \(\beta ^{(n)}(\xi )\)  and \(\alpha ^{(n)}(\xi )\)  into sums analogous to that for \(b(x)\)  at (REF ). We derive first expressions for the components of \(\beta ^{(n)}(\xi )\) . Recall that \(\xi =(n^S, n_0^I, n_1^I,\dots ,n^W)\) , \(n^I=\sum _{k=0}^{\infty } n_k^I\)  and \(n^E = \sum _{k=0}^{\infty } k n_k^I\) . Then
\(\beta ^{(n)}_R(\xi )&=n^{-1}\gamma (0, -n^I, -n^E,0) =b_R(x(\xi )),\nonumber \\\beta ^{(n)}_D(\xi )&=\omega (1-\alpha ) n^E n^{-1}(0, 0, -1, 0)=b_D(x(\xi )),\nonumber \\\beta ^{(n)}_{RWR}(\xi )&=\omega \alpha n^E \left(\frac{n-n^S-n^I-2}{n-2}\right) n^{-1}(0,0,-1,0)\nonumber \\&=\omega \alpha x^3(\xi )\left(\frac{n(1-x^1(\xi )-x^2(\xi ))-2}{n-2}\right)(0,0,-1,0),\nonumber \\\beta ^{(n)}_{RWS}(\xi )&=n^E \omega \alpha \frac{n^S-1}{n-2} n^{-1}(0,0,-1,0)=\omega \alpha x^3(\xi )\frac{nx^1(\xi ))-1}{n-2} (0,0,-1,0),\nonumber \\\beta ^{(n)}_{I}(\xi )&=\lambda n^E n^{-1}\left(-1,1,\frac{\mu _n(n^S-1)}{n}-1+2\frac{n^W}{n^S}-\frac{n^E-1}{n^S}, -2\frac{n^W}{n^S}\right)\\&=\lambda x^3(\xi )\left(-1,1,\mu _n(x^1(\xi )-n^{-1})-1+2\frac{x^4(\xi )}{x^1(\xi )}-\frac{x^3(\xi )-n^{-1}}{x^1(\xi )}, -2\frac{x^4(\xi )}{x^1(\xi )}\right).\nonumber \) 
